The schematic's base has a 5 sided polygon labeled "IN". What does it represent?
That is the input to the circuit.
Most circuit diagrams don't show the whole circuit, just the part that is of interest. This circuit doesn't show the power supply, or what drives the input. Here the input is going to be a voltage above about 3 V relative to ground to turn the transistor on, and near 0 V to turn the transistor off.
Why is a resistor @ 560 ohms there? I often see a resistor attached to a transistor and I wonder why.
The voltage between the base and emitter can't exceed about 0.7V. The resistor is there to limit the current. If the input is 3 V, with 0.7 V on the base, there will be 2.3 V across the resistor so a current of around 4 mA will flow.
The collector is attached to ground.
It's not the collector. It's the emitter.
What does ground represent? A clamp on a rod pounded into the ground?
It could be, but here it just represents the point against which other voltages are measured. If it were on a car, it would probably be the body of the car.
On an architect's drawing, there's often a point that is the reference against which heights are measure. The ground point is a bit like that for voltage.
The emitter has 5 volt positive lead connected to an inductor? with a diode. The box is a controller? What is this thing?
The box is a relay. A relay is an electromagnet, which has considerable inductance, but its purpose is to operate a switch. That is usually called the coil. The outline of the box sort of indicates the connection between the coil between terminals 2 and 5 and the switch which is between terminals 1, 3 and 4. Terminals 1 and 4 are connected by the relay when the coil has no current. Terminals 1 and 3 are connected by the relay when the coil has current flowing.
The circuit doesn't show what is connected to those terminals of the relay. There would usually be something, as that is usually the purpose of a relay.
The relay coil has inductance. When the transistor turns off, the current flowing in the collector of the transistor can change to zero very quickly. The inductance of the relay coil can cause a momentary voltage spike which can be very large. The diode allows the current in the relay to change much more slowly, reducing the voltage spike.
https://en.wikipedia.org/wiki/Flyback_diode has more information about how a diode can be connected like that.